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The Wonders

Planiver se

"Planiversal scientists are not a very common breed."

-Alexander Keewatin Dewdney

s far as anyone knows the only existing universe is the one

we live in, with its three dimensions of space and one of time.

.It is not hard to imagine, as many science-fiction writers have,

that intelligent organisms could live in a fourdimensional space,

but two dimensions offer such limited degrees of freedom that it has

long been assumed intelligent two-space life forms could not exist.

Two notable attempts have nonetheless been made to describe such

organisms.

T h e L a s t R e c r e a t i o n s

In 1884 Edwin Abbott Abbott, a London clergyman, published his

satirical novel Flatland. Unfortunately the book leaves the reader almost

entirely in the dark about Flatland's physical laws and the technology

developed by its inhabitants, but the situation was greatly improved in

1907 when Charles Howard Hinton published An Episode of Flatland. Although written in a flat style and with cardboard characters, Hinton's

story provided the first glimpses of the possible science and technology

of the two-dimensional world. His eccentric book is, alas, long out of

print, but you can read about it in the chapter "Flatlands" in my book

The Unexpected Hanging and Other Mathematical Diversions (Simon &

Schuster, 1969).

In "Flatlands" I wrote: "It is amusing to speculate on d i m e n -

sional physics and the kinds of simple mechanical devices that would

be feasible in a flat world." This remark caught the attention of Alexander

Keewatin Dewdney, a computer scientist at the University of Western

Ontario. Some of his early speculations on the subject were set down in

1978 in a university report and in 1979 in "Exploring the Planiverse,"

an article in Journal of Recreational Mathematics (Vol. 12, No. 1, pages

16-20; September). Later in 1979 Dewdney also privately published

"Two-dimensional Science and Technology," a 97-page tour de force. It

is hard to believe, but Dewdney actually lays the groundwork for what

he calls a planiverse: a possible two-dimensional world. Complete with

its own laws of chemistry, physics, astronomy, and biology, the planiverse

is closely analogous to our own universe (which he calls the steriverse)

and is apparently free of contradictions. I should add that this remark-

able achievement is an amusing hobby for a mathematician whose seri-

ous contributions have appeared in some 30 papers in technical jour-

nals.

Dewdney's planiverse resembles Hinton's in having an earth that

he calls (as Hinton did) Astria. Astria is a disklike planet that rotates in

planar space. The Astrians, walking upright on the rim of the planet,

T h e Wonde rs o f a P lan iverse 3

can distinguish east and west and up and down. Naturally there is no

north or south. The "axis" of Astria is a point at the center of the

circular planet. You can think of such a flat planet as being truly two-

dimensional or you can give it a very slight thickness and imagine it as

moving between two frictionless planes.

As in our world, gravity in a planiverse is a force between objects

that varies directly with the product of their masses, but it varies in-

versely with the linear distance between them, not with the square of

that distance. O n the assumption that forces such as light and gravity in

a planiverse move in straight lines, it is easy to see that the intensity of

such forces must vary inversely with linear distance. The familiar t e s

book figure demonstrating that in our world the intensity of light varies

inversely with the square of distance is shown at the top of Figure 1. The obvious planar analogue is shown at the bottom of the illustration.

Figure 1 NINE

FOUR SQUARE

ONE

THREE INCHES

I

4 The L a s t R e c r e a t i o n s

To keep his whimsical project from "degenerating into idle specula-

tion" Dewdney adopts two basic principles. The "principle of similar-

ity" states that the planiverse must be as much like the steriverse as

possible: a motion not influenced by outside forces follows a straight

line, the flat analogue of a sphere is a circle, and so on. The "principle

of modification" states that in those cases where one is forced to choose

between conflicting hypotheses, each one equally similar to a steriversal

theory, the more fundamental one must be chosen and the other modi-

fied. To determine which hypothesis is more fundamental Dewdney

relies on the hierarchy in which physics is more fundamental than

chemistry, chemistry more fundamental than biology, and so on.

To illustrate the interplay between levels of theory Dewdney consid-

ers the evolution of the planiversal hoist in Figure 2. The engineer who

designed it first gave it arms thinner than those in the illustration, but

when a metallurgist pointed out that planar materials fracture more

easily than their three-space counterparts, the engineer made the arms

Figure 2

The W o n d e r s o f a P l a n i v e r s e

thicker. Later a theoretical chemist, invoking the principles of similarity

and modification at a deeper level, calculated that the planiversal m e

lecular forces are much stronger than had been suspected, and so the

engineer went back to thinner arms.

The principle of similarity leads Dewdney to posit that the planiverse

is a threedimensional continuum of space-time containing matter com-

posed of molecules, atoms, and fundamental particles. Energy is propa-

gated by waves, and it is quantized. Light exists in all its wavelengths

and is refracted by planar lenses, making possible planiversal eyes,

planiversal telescopes, and planiversal microscopes. The planiverse shares

with the steriverse such basic precepts as causality; the first and second

laws of thermodynamics; and laws concerning inertia, work, friction,

magnetism, and elasticity.

Dewdney assumes that his planiverse began with a big bang and is

currently expanding. An elementary calculation based on the inverse-

linear gravity law shows that regardless of the amount of mass in the

planiverse the expansion must eventually halt, so that a contracting

phase will begin. The Astrian night sky will of course be a semicircle

along which are scattered twinkling points of light. If the stars have

proper motions, they will continually be occulting one another. If Astria

has a sister planet, it will over a period of time occult every star in the

sky* We can assume that Astria revolves around a sun and rotates, thereby

creating day and night. In a planiverse, Dewdney discovered, the only

stable orbit that continually retraces the same path is a perfect circle.

Other stable orbits roughly elliptical in shape are possible, but the axis

of the ellipse rotates in such a way that the orbit never exactly closes.

Whether planiversal gravity would allow a moon to have a stable orbit

around Astria remains to be determined. The difficulty is due to the

sun's gravity, and resolving the question calls for work on the planar

analogue of what our astronomers know as the three-body problem.

6 T h e L a s t R e c r e a t i o n s

Dewdney analyzes in detail the nature of Astrian weather, using

analogies to our seasons, winds, clouds, and rain. An Astrian river

would be indistinguishable from a lake except that it might have

faster currents. One peculiar feature of Astrian geology is that water

cannot flow around a rock as it does on the earth. As a result

rainwater steadily accumulates behind any rock on a slope, tending

to push the rock downhill: the gentler the slope is, the more water

accumulates and the stronger the push is. Dewdney concludes that

given periodic rainfall the Astrian surface would be unusually flat

and uniform. Another consequence of the inability of water to move

sideways on Astria is that it would become trapped in pockets within

the soil, tending to create large areas of treacherous quicksand in the

hollows of the planet. One hopes, Dewdney writes, that rainfall is

infrequent on Astria. Wind too would have much severer effects on

Astria than on the earth because like rain it cannot "go around"

objects.

Dewdney devotes many pages to constructing a plausible chemistry

for his planiverse, modeling it as much as possible on three-dimen-

sional matter and the laws of quantum mechanics. Figure 3 shows

Dewdney's periodic table for the first 16 planiversal elements. Because

the first two are so much like their counterparts in our world, they are

called hydrogen and helium. The next 10 have composite names to

suggest the steriversal elements they most resemble; for example, lithrogen

combines the properties of lithium and nitrogen. The next four are

named after Hinton, Abbott, and the young lovers in Hinton's novel,

Harold Wall and Laura Cartwright.

In the flat world atoms combine naturally to form molecules, but of

course only bonding that can be diagrammed by a planar graph is

allowed. (This result follows by analogy from the fact that intersecting

bonds do not exist in steriversal chemistry.) As in our world, two asym-

metric molecules can be mirror images of each other, so that neither

The Wonders o f a Planiverse

ATOMIC NUMBER

HYDROGEN

HELIUM

NEOCARBON Nc

SODALINUM I Sa 1 MAGNILICON I Mc

ALUPHORUS ( Ap

SULFICON I Sp

CHLOPHORUS Cp

ARGOFUR I If HlNTONlUM Hn 1 ABBOGEN I Ab HAROLDIUM Wa

SHELL STRUCTURE VALENCE 1s 2s 2p 3s 3p 3d 4s 4p . . .

Figure 3

one can be "turned over" to become identical with the other. There are

striking parallels between planiversal chemistry and the behavior of

steriversal monolayers on crystal surfaces [see ''Two-dimensional Mat-

ter," by J. G. Dash; Scientific American, May 19731. In our world mol-

ecules can form 230 distinct crystallographic groups, but in the planiverse

they can form only 17. I am obliged to pass over Dewdney's specula-

tions about the diffusion of molecules, electrical and magnetic laws,

analogues of Maxwell's equations, and other subjects too technical to

summarize here.

Dewdney assumes that animals on Astria are composed of cells that

cluster to form bones, muscles, and connective tissues similar to those

found in steriversal biology. He has little difficulty showing how these

bones and muscles can be structured to move appendages in such a way

that the animals can crawl, walk, fly, and swim. Indeed, some of these

movements are easier in a planiverse than in our world. For example, a

steriversal animal with two legs has considerable difficulty balancing

while walking, whereas in the planiverse if an animal has both legs on

8 The L a s t R e c r e a t i o n s

the ground, there is no way it can fall over. Moreover, a flying planiversal

animal cannot have wings and does not need them to fly; if the body of

the animal is aerodynamically shaped, it can act as a wing (since air can

go around it only in the plane). The flying animal could be propelled by

a flapping tail.

Calculations also show that Astrian animals probably have much

lower metabolic rates than terrestrial animals because relatively little

heat is lost through the perimeter of their body. Furthermore, animal

bones can be thinner on Astria than they are on the earth, because

they have less weight to support. Of course, no Astrian animal can have

an open tube extending from its mouth to its anus, because if it did, it

would be cut in two.

In the appendix to his book The Structure and Evolution of the

Universe (Harper, 1959) G. J. Whitrow argues that intelligence could

not evolve in two-space because of the severe restrictions two dimen-

sions impose on nerve connections. "In three or more dimensions," he

writes, "any number of [nerve] cells can be connected with [one an- )

other] in pairs without intersection of the joins, but in two dimensions

the maximum number of cells for which this is possible is only four."

Dewdney easily demolishes this argument, pointing out that if nerve

cells are allowed to fire nerve impulses through "crossover points," they

can form flat networks as complex as any in the steriverse. Planiversal

minds would operate more slowly than steriversal ones, however, be-

cause in the two-dimensional networks the pulses would encounter

more interruptions. (There are comparable results in the theory of two-

dimensional automatons.)

Dewdney sketches in detail the anatomy of an Astrian female fish

with a sac of unfertilized eggs between its two tail muscles. The fish has

an external skeleton, and nourishment is provided by the internal circu-

lation of food vesicles. If a cell is isolated, food enters it through a

membrane that can have only one opening at a time. If the cell is in

T h e Wonders o f a Planiverse 9

contact with other cells, as in a tissue, it can have more than one

opening at a time because the surrounding cells are able to keep it

intact. We can of course see every internal organ of the fish or of any

other planiversal life form, just as a four-dimensional animal could see

all our internal organs.

Dewdney follows Hinton in depicting his Astrian people schemati-

cally, as triangles with two arms and two legs. Hinton's Astrians, how-

ever, always face in the same direction: males to the east and females to

the west. In both sexes the arms are on the front side, and there is a

single eye at the top of the triangle, as shown in Figure 4. Dewdney's

Astrians are bilaterally symmetrical, with an arm, a leg, and an eye on

each side, as shown in the illustration's center. Hence these Astrians,

like terrestrial birds or horses, can see in opposite directions. Naturally

the only way for one Astrian to pass another is to crawl or leap over

him. My conception of an Astrian bug-eyed monster is shown at the

right in the illustration. This creature's appendages serve as either arms

or legs, depending on which way it is facing, and its two eyes provide

binocular vision. With only one eye an Astrian would have a largely

onedimensional visual world, giving him a rather narrow perception of

reality. O n the other hand, parts of objects in the planiverse might be

distinguished by their color, and an illusion of depth might be created

by the focusing of the lens of the eye.

O n Astria building a house or mowing a lawn requires less work

Figure 4


than it does on the earth because the amount of material involved is

considerably smaller. As Dewdney points out, however, there are still

formidable problems to be dealt with in a d i m e n s i o n a l world: "As-

suming that the surface of the planet is absolutely essential to support

life-giving plants and animals, it is clear that very little of the Astrian

surface can be disturbed without inviting the biological destruction of

the planet. For example, here on earth we may build a modest highway

through the middle of several acres of rich farmland and destroy no

more than a small percentage of it. A corresponding highway on Astria

with destroy all the 'acreage' it passes over. . . . Similarly, extensive cities

would quickly use up the Astrian countryside. It would seem that the

only alternative for the Astrian technological society is to go under-

ground." A typical subterranean house with a living room, two bed-

rooms, and a storage room is shown in Figure 5 . Collapsible chairs and

tables are stored in recesses in the floors to make the rooms easier to

walk through.

The many simple threedimensional mechanical elements that have

obvious analogues on Astria include rods, levers, inclined planes, springs,

hinges, ropes, and cables (see Figure 6, top). Wheels can be rolled

along the ground, but there is no way to turn them on a fixed axle.

Screws are impossible. Ropes cannot be knotted; but by the same to-

ken, they never tangle. Tubes and pipes must have partitions, to keep

their sides in place, and the partitions have to be opened (but never all

of them at once) to allow anything to pass through. It is remarkable that

in spite of these severe constraints many flat mechanical devices can be

built that will work. A faucet designed by Dewdney is shown in Figure

6, bottom. To operate it the handle is lifted. This action pulls the valve

away from the wall of the spout, allowing the water to flow out. When

the handle is released, the spring pushes the valve back.

The device shown in Figure 7 serves to open and close a door (or

a wall). Pulling down the lever at the right forces the wedge at the

T h e W o n d e r s o f a P l a n i v e r s e

Figure 5

1 2 T h e L a s t R e c r e a t i o n s

ROD SPRING HINGE

Figure 6 (top)

-- -- - - - -

Figure 6 (bottom)

bottom to the left, thereby allowing the door to swing upward (carrying

the wedge and the levers with it) on a hinge at the top. The door is

opened from the left by pushing up on the other lever. The door can be

lowered from either side and the wedge moved back to stabilize the wall

by moving a lever in the appropriate direction. This device and the

faucet are both mechanisms with permanent planiversal hinges: circu-

lar knobs that rotate inside hollows but cannot be removed from them.

Figure 8 depicts a planiversal steam engine whose operation paral-

lels that of a steriversal engine. Steam under pressure is admitted into

the cylinder of the engine through a sliding valve that forms one of its

walls (top). The steam pressure causes a piston to move to the right until

steam can escape into a reservoir chamber above it. The subsequent

loss of pressure allows the compound leaf spring at the right of the

cylinder to drive the piston back to the left (bottom). The sliding valve is

The Wonders o f a Planiverse 1 3

Figure 7

closed as the steam escapes into the reservoir, but as the piston moves

back it reopens, pulled to the right by a spring-loaded arm.

Figure 9 depicts Dewdney's ingenious mechanism for unlocking a

door with a key. This planiversal lock consists of three slotted tumblers

(a) that line up when a key is inserted (b) so that their lower halves

move as a unit when the key is pushed (c). The pushing of the key is

transmitted through a lever arm to the master latch, which pushes

The L a s t R e c r e a t i o n s

Figure 8

down on a slave latch until the door is free to swing to the right (d). The

bar on the lever arm and the lip on the slave latch make the lock

difficult to pick. Simple and compound leaf springs serve to return all

the parts of the lock except the lever arm to their original positions

when the door is opened and the key is removed. When the door

closes, it strikes the bar on the lever arm, thereby returning that piece to

its original position as well. This flat lock could actually be employed in

the steriverse; one simply inserts a key without twisting it.

"It is amusing to think," writes Dewdney, "that the rather exotic

design pressures created by the planiversal environment could cause us

to think about mechanisms in such a different way that entirely novel

solutions to old problems arise. The resulting designs, if steriversally

practical, are invariably ~pace~saving."

T h e W o n d e r s of a P l an ive r s e

Figure 8 (continued)

Thousands of challenging planiversal problems remain unsolved.

Is there a way, Dewdney wonders, to design a two-dimensional

windup motor with flat springs or rubber bands that would store

energy? What is the most efficient design for a planiversal clock,

telephone, book, typewriter, car, elevator or computer? Will some

machines need a substitute for the wheel and axle? Will some need

electric power?

There is a curious pleasure in trying to invent machines for what

Dewdney calls "a universe both similar to and yet strangely different

from ours." As he puts it, "from a small number of assumptions many

phenomena seem to unfurl, giving one the sense of a kind of separate

existence of this twodimensional world. One finds oneself speaking,

willy-nilly, of the planiverse as opposed to a planiverse. . . . [For] those

The L a s t R e c r e a t i o n s

Figure 9

who engage in it positively, there is a kind of strange enjoyment, like

[that of] an explorer who enters a land where his own perceptions play

a major role in the landscape that greets his eyes."

Some philosophical aspects of this exploration are not trivial. In

constructing a planiverse one sees immediately that it cannot be built

without a host of axioms that Leibniz called the "compossible" ele-

ments of any possible world, elements that allow a logically consistent

structure. Yet as Dewdney points out, science in our universe is based

mainly on observations and experiments, and it is not easy to find any

T h e Wonders o f a Planiverse

underlying axioms. In constructing a planiverse we have nothing to

observe. We can only perform gedanken experiments (thought experi-

ments) about what might be observed. "The experimentalist's loss,"

observes Dewdney, "is the theoretician's gain."

A marvelous exhibit could be put on of working models of planiversal

machines, cut out of cardboard or sheet metal, and displayed on a

surface that slopes to simulate planiversal gravity. One can also imagine

beautiful cardboard exhibits of planiversal landscapes, cities, and houses.

Dewdney has opened up a new game that demands knowledge of both

science and mathematics: the exploration of a vast fantasy world about

which at present almost nothing is known.

It occurs to me that Astrians would be able to play two-dimen-

sional board games but that such games would be as awkward for them

as three-dimensional board games are for us. I imagine them, then,

playing a variety of linear games on the analogue of our 8-by8 chess-

board. Several games of this type are shown in Figure 10. Part a shows

the start of a checkers game. Pieces move forward only, one cell at a

time, and jumps are compulsory. The linear game is equivalent to a

game of regular checkers with play confined to the main diagonal of a

standard board. It is easy to see how the second player wins in ratio-

nal play and how in misere, or "giveaway," checkers the first player

wins just as easily. Linear checkers games become progressively

harder to analyze as longer boards are introduced. For example,

which player wins standard linear checkers on the 11-cell board

when each player starts with checkers on the first four cells at his

end of the board?

Part b in the illustration shows an amusing Astrian analogue of

chess. O n a linear board a bishop is meaningless and a queen is the

same as a rook, so the pieces are limited to kings, knights, and rooks.

The only rule modification needed is that a knight moves two cells in

either direction and can jump an intervening piece of either color. If


I I I I I I 1 e t I I I I I I I j

Figure 10

the game is played rationally, will either White or Black win or will

the game end in a draw? The question is surprisingly tricky to an-

swer.

Linear go, played on the same board, is by no means trivial. The

version I shall describe was invented 10 years ago by James Marston

Henle, a mathematician who is now at Smith College. Called pinch by

Henle, it is published here for the first time.

In the4game of pinch players take turns placing black and white

stones on the cells of the linear board, and whenever the stones of one

player surround the stones of the other, the surrounded stones are

removed. For example, both sets of white stones shown in part c of

Figure 10 are surrounded. Pinch is played according to the following

two rules.

Rule 1: No stone can be placed on a cell where it is surrounded

unless that move serves to surround a set of enemy stones. Hence in

the situation shown in part d of the illustration White cannot play on

The Wonders o f a Planiverse 19

cells 1, 3, or 8, but he can play on cell 6 because this move serves to

surround cell 5. Rule 2: A stone cannot be placed on a cell from which a stone was

removed on the last play if the purpose of the move is to surround

something. A player must wait at least one turn before making such a

move. For example, in part e of the illustration assume that Black plays

on cell 3 and removes the white stones on cells 4 and 5. White cannot

play on cell 4 (to surround cell 3) for his next move, but he may do so

for any later move. He can play on cell 5, however, because even though

a stone was just removed from that cell, the move does not serve to

surround anything. This rule is designed to decrease the number of

stalemates, as is the similar rule in go.

Two-cell pinch is a trivial win for the second player. The three- and

four-cell games are easy wins for the first player if he takes the center in

the three-cell game and one of the two central cells in the four-cell one.

The five-cell game is won by the second player and the six- and seven*

cell games are won by the first player. The eight-cell game jumps to such

a high level of complexity that it becomes very exciting to play. Fortunes

often change rapidly, and in most situations the winning player has

only one winning move.

Answers

In 1 l-cell linear checkers (beginning with Black on cells 1,2,3, and

4 and White on cells 8, 9, 10, and 11) the first two moves are forced:

Black to 5 and White to 7. To avoid losing, Black then goes to 4, and

White must respond by moving to 8. Black is then forced to 3 and

White to 9. At this point Black loses with a move to 2 but wins with a

move to 6. In the latter case White jumps to 5, and then Black jumps

to 6 for an easy end-game victory.

On the eight-cell linear chessboard White can win in at most six

moves. Of White’s four opening moves, R×R is an instant stalemate andthe shortest possible game. R-5 is a quick loss for White if Black playsR×R. Here White must respond with N-4, and then Black mates on hissecond move with R×N. This game is one of the two “fool’s mates,” orshortest possible wins. The R-4 opening allows Black to mate on his sec-ond or third move if he responds with N-5.

White’s only winning opening is N-4. Here Black has three possiblereplies:

1. R×N.

In this case White wins in two moves with R×R.

2. R-5.

White wins with K-2. If Black plays R-6, White mates with N×R. IfBlack takes the knight, White takes the rook, Black moves N-5, andWhite mates by taking Black’s knight

3. N-5.

This move delays Black’s defeat the longest. In order to win Whitemust check with N×R, forcing Black’s king to 7. White moves his rookto 4. If Black plays K×N, White’s king goes to 2, Black’s K-7 is forced,and White’s R×N wins. If Black plays N-3 (check), White moves theking to 2. Black can move only the knight. If he plays N-1, White mateswith N-8. If Black plays N-5, White’s N-8 forces Black’s K×N, and thenWhite mates with R×N.

The first player also has the win in eight-cell pinch (linear go) byopening on the second cell from an end, a move that also wins the sixand seven-cell games. Assume that the first player plays on cell 2. Hisunique winning responses to his opponent’s plays on 3, 4, 5, 6, 7, and 8are respectively 5, 7, 7, 7, 5, and 6. I leave the rest of the game to thereader. It is not known whether there are other winning opening moves.James Henle, the inventor of pinch, assures me that the second player

20 The Last Rec reat ions

T h e W o n d e r s o f a P l a n i v e r s e

wins the nine-cell game. H e has not tried to analyze boards with more

than nine cells.

My column on the planiverse generated enormous interest. Dewdney

received some thousand letters offering suggestions about flatland sci-

ence and technology. In 1979 he privately printed Two-Dimensional Sci-

ence and Technology, a monograph discussing these new results. Two years

later he edited another monograph, A Symposium of Two-Dimensional Sci-

ence and Technology. It contained papers by noted scientists, mathemati-

cians, and laymen, grouped under the categories of physics, chemistry,

astronomy, biology, and technology. Newsweek covered these monographs

in a two-page article, "Life in Two Dimensions" (January 18, 1980), and

a similar article, "Scientific Dreamers' Worldwide Cult," ran in Canada's

Maclean's magazine (January 11, 1982). Omni (March 1983), in an article

on "Flatland Redux," included a photograph of Dewdney shaking hands

with an Astrian.

In 1984 Dewdney pulled it all together in a marvelous work, half nonfic-

tion and half fantasy, titled The Planiverse and published by Poseidon Press,

an imprint of Simon 6;c Schuster. That same year he took over the mathemat-

ics column in Scientific American, shifiing its emphasis to computer recre-

ations. Several collections of his columns have been published by W. H. Freeman: The Armchair Universe (1987), The Turing Omnibus (1989), and The

Magic Machine (1990).

An active branch of physics is now devoted to planar phenomena. It

involves research on the properties of surfaces covered by a film one molecule

thick, and a variety of two-dimensional electrostatic and electronic effects.

Exploring possible flatlands also relates to a philosophical fad called "pos-

sible worlds." Extreme proponents of this movement actually argue that if a

universe is logically possible-that is, free of logical contradictions-it is just as

"real" as the universe in which we flourish.

In Childhood's End Arthur Clarke describes a giant planet where intense


gravity has forced life to evolve almost flat forms with a vertical thickness of

one centimeter.

The following letter from J. Richard Gott 111, an astrophysicist at Princeton

University, was published in Scientific American (October 1980):

I was interested in Martin Gardner's article on the physics of Flatland, because for some years I have given the students in my general relativity class the problem of deriving the theory of general relativity for Flatland. The results are surprising. One does not obtain the Flatland analogue of Newtonian theory (masses with gravitational fields falling off like l / r ) as the weak-field limit. General relativity in Flatland predicts no gravitational waves and no action at a distance. A planet in Flatland would produce no gravitational effects beyond its own radius. In our four-dimensional space-time the energy momentum tensor has 10 independent components, whereas the Riemann curvature tensor has 20 independent components. Thus it is possible to find solutions to the vacuum field equations G,,, = 0 (where all components of the energy momentum tensor are zero) that have a nonzero curvature. Black-hole solutions and the gravitational-field solution external to a planet are examples. This allows gravitational waves and action at a distance. Flatland has a three-dimensional space- time where the energy momentum tensor has six independent components and the Riemann curvature tensor also has only six independent components. In the vacuum where all components of the energy momentum tensor are zero all the components of the Riemann curvature tensor must also be zero. No action at a distance or gravity waves are allowed.

Electromagnetism in Flatland, o n the other hand, be- haves just as one would expect. The electromagnetic field tensor in four-dimensional space-time has six independent components that can be expressed as vector E and B fields with three components each. The electromagnetic field tensor in a three-dimensional space-time (Flatland) has three independent components: a vector E field with two compo-


nents and a scalar B field. Electromagnetic radiation exists, and charges have electric fields that fall off like l/r.

Two more letters, published in the same issue, follow. John S. Harris, of

Brigham Young University's English Department, wrote:

As I examined Alexander Keewatin Dewdney's planiversal devices in Martin Gardner's article o n science and technology in a two-dimensional universe, I was struck with the similarity of the mechanisms to the lockwork of the Mauser military pistol of 1895. This remarkable automatic pistol (which had many later variants) had no pivot pins or screws in its func- tional parts. Its entire operation was through sliding cam surfaces and two-dimensional sockets (called hinges by Dewdney). In- deed, the lockwork of a great many firearms, particularly those of the 19th century, follows essentially planiversal principles. For examples see the cutaway drawings in Book of Pistols and Revolvers by W. H . B. Smith.

Gardner suggests an exhibit of machines cut from cardboard, and that is exactly how the firearms genius John Browning worked. He would sketch the parts of a gun o n paper or cardboard, cut out the individual parts with scissors (he often carried a small pair in his vest pocket), and then would say to his brother Ed, "Make me a part like this." Ed would ask, "How thick, John?" John would show a dimension with his thumb and forefinger, and Ed would measure the distance with calipers and make the part. The result is that virtually every part of the 100 or so Browning designs is essentially a two-dimensional shape with an added thickness.

This planiversality of Browning designs is the reason for the obsolescence of most of them. Dewdney says in his en- thusiasm for the planiverse that "such devices are invariably space-saving." They are also expensive to manufacture. The Browning designs had to be manufactured by profiling machines: cam-following vertical milling machines. In cost of manufacture such designs cannot compete with designs that can be produced by automatic screw-cutting lathes, by broach-


ing machines, by stamping, or by investment casting. Thus although the Browning designs have a marvelous aesthetic appeal, and although they function with delightful smooth- ness, they have nearly all gone out of production. They simply got too expensive to make.

Stefan Drobot, a mathematician at Ohio State University, had this

to say:

In Martin Gardner's article he and the authors he quotes seem to have overlooked the following aspect of a "planiverse": any communication by means of a wave process, acoustic or electromagnetic, would in such a universe be impossible. This is a consequence of the Huygens principle, which expresses a mathematical property of the (fundamental) solutions of the wave equation. More specifically, a sharp impulse-type signal (represented by a "delta function") originating from some point is propagated in a space of three spatial dimensions in a manner essentially different from that in which it is propagated in a space of two spatial dimensions. In three-dimensional space the signal is propagated as a sharp-edged spherical wave without any trail. This property makes it possible to communicate by a wave process because two signals following each other in a short time can be distinguished.

In a space with two spatial dimensions, on the other hand, the fundamental solution of the wave equation represents a wave that, although it too has a sharp edge, has a trail of theoretically infinite length. An observer at a fixed distance from the source of the signal would perceive the oncoming front (sound, light, etc.) and then would keep perceiving it, although the intensity would decrease in time. This fact would make communication by any wave process impossible because it would not allow two signals following each other to be distinguished. More practically such communication would take much more time. This letter could not be read in the planiverse, although it is (almost) two-dimensional.

My linear checkers and chess prompted many interesting letters. Abe


Schwartz assured me that on the 11-cell checker field Black also wins if the

game is give-away. I. Richard Lapidus suggested modifying linear chess by

interchanging knight and rook (the game is a draw), by adding more cells,

by adding pawns that capture by moving forward one space, or by combi-

nations of the three modifications. If the board is long enough, he sug

gested duplicating the pieces-two knights, two rooks-and adding several

pawns, allowing a pawn a two-cell start option as in standard chess. Peter

Stampolis proposed sustituting for the knight two pieces called "kops" be-

cause they combine features of knight and bishop moves. One kop moves

only on white cells, the other only on black.

Of course many other board games lend themselves to linear forms, for

example, Reversi (also called Othello), or John Conway's Phutball, described

in the two-volume Winning Ways written by Elwyn Berlekamp, Richard Guy,

and John Conway.

References

AN EPISODE OF FLATLAND. C. H. Hinton. Swan Sonnenschein & Co., 1907.

FLATLAND: A ROMANCE OF MANY DIMENSIONS, BY A. SQUARE. E. A. Abbott.

Dover Publications, Inc., 1952. Several other paperback editions are cur-

rently in print.

SPHERELAND: A FANTASY ABOUT CURVED SPACES AND AN EXPANDING UNIVERSE.

Thomas Y. Crowell, 1965.

THE PLANIVERSE. A. K. Dewdney. Poseidon, 1984.

ALLEGORY THROUGH THE COMPUTER CLASS: SUFISM IN DEWDNEY'S PLANIVERSE.

P. J. Stewart in Sufi, Issue 9, pages 26-30; Spring 1991.

200 PERCENT OF NOTHING: AN EYE OPENING TOUR THROUGH THE TWISTS AND

TURNS OF MATH ABUSE AND INNUMERACY. A. K. Dewdney. Wiley, 1994.

INTRODUCTORY COMPUTER SCIENCE: BITS OF THEORY, BYTES OF PRACTICE. A. K. Dewdney. Freeman, 1996.

the last recreations: hydras, eggs, and other … written in a flat style and with cardboard...

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